Monoidal structure on a category with products and with terminal object - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:54:38Z http://mathoverflow.net/feeds/question/111193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111193/monoidal-structure-on-a-category-with-products-and-with-terminal-object Monoidal structure on a category with products and with terminal object Sergei Akbarov 2012-11-01T19:13:01Z 2012-11-02T09:24:09Z <p>Let $K$ be a category with products $(X,Y)\mapsto X\sqcap Y$ and with a terminal object $T$. It seems obvious to me that $\sqcap$ and $T$ define a structure of a monoidal category on $K$, but I can't find a reference. When I try to prove this myself I come to amazingly bulky constructions. Is there a text where this is accurately proved, or at least formulated? </p> http://mathoverflow.net/questions/111193/monoidal-structure-on-a-category-with-products-and-with-terminal-object/111202#111202 Answer by Wouter Stekelenburg for Monoidal structure on a category with products and with terminal object Wouter Stekelenburg 2012-11-01T19:57:58Z 2012-11-01T19:57:58Z <p>You can find it as an example of a monoidal category in Tom Leinster's "Higher Operads, Higher Categories", which contains loads of coherence proofs for higher categories.</p> http://mathoverflow.net/questions/111193/monoidal-structure-on-a-category-with-products-and-with-terminal-object/111223#111223 Answer by David Roberts for Monoidal structure on a category with products and with terminal object David Roberts 2012-11-01T22:53:36Z 2012-11-01T22:53:36Z <p>You need to have chosen products for every pair of objects using the axiom of choice, otherwise you don't get a product functor, just a product <em><a href="http://ncatlab.org/nlab/show/anafunctor" rel="nofollow">anafunctor</a></em> and I recommend you don't try to use those just yet. Then you can prove - and this is the key step - that any two bracketings of an iterated product are isomorphic in a unique way when you demand the isomorphism respects the all the projections. The unique such isomorphism for a triple product is then the associator, and the uniqueness of the isomorphism for the 4-fold product means that the pentagon commutes. Ditto with the other coherence conditions.</p> <p>Note that you can choose the product of any object $X$ with the terminal object to be $X$. This makes the unit conditions automatic.</p> http://mathoverflow.net/questions/111193/monoidal-structure-on-a-category-with-products-and-with-terminal-object/111251#111251 Answer by Buschi Sergio for Monoidal structure on a category with products and with terminal object Buschi Sergio 2012-11-02T07:26:39Z 2012-11-02T09:24:09Z <p>Is very easy prove that $(Set, \times, 1)$ is monoidal (by elements checking). Now let $\mathcal{C}$ a category by finite product $\times$ and (then) with a final object $1$. Consider the axioms of monoidal category for $(\mathcal{C}, \times , 1)$ stated by diagrams (see for example p.462 of "Closed Categories" by Eilenberg &amp; Kelly, LA Jolla 1967), now it remains to prove that these diagrams are commutative. COnsider a such diagram $\textbf{D}$ and a (general) object $X\in \mathcal{C}$ and the representable $(X, -): \mathcal{C}\to Set: A \mapsto (X, A)$, acting by $(X, -)$ on this diagram, we get a similar diagram in $Set$, say $X(\textbf{D})$, and $(X, -)$ preserve the product $\times$ and the final object $1$, now we just know that $(Set, \times, 1)$ is monoidal, then $X(\textbf{D})$ is commutative. Because this is true for each object $X$, by Yoneda lemma follow that $\textbf{D}$ is commutative (more easily observe that given $f, g: A \to B$, if $(X, f)=(X, g): (X, A)\to (X, B)$ for each $X$ then $f=g$ (consider $X=A$ and $1_A$)).</p>